Alexander Domahidi

Efficient interior point methods for multistage problems arising in receding horizon control

Receding horizon control requires the solution of an optimization problem at every sampling instant. We present efficient interior point methods tailored to convex multistage problems, a problem class which most relevant MPC problems with linear dynamics can be cast in, and specify important algorithmic details required for a high speed implementation with superior numerical stability. In particular, the presented approach allows for quadratic constraints, which is not supported by existing fast MPC solvers. A categorization of widely used MPC problem formulations into classes of different complexity is given, and we show how the computational burden of certain quadratic or linear constraints can be decreased by a low rank matrix forward substitution scheme. Implementation details are provided that are crucial to obtain high speed solvers.We present extensive numerical studies for the proposed methods and compare our solver to three well-known solver packages, outperforming the fastest of these by a factor 2-5 in speed and 3-70 in code size. Moreover, our solver is shown to be very efficient for large problem sizes and for quadratically constrained QPs, extending the set of systems amenable to advanced MPC formulations on low-cost embedded hardware.

High-Bandwidth Explicit Model Predictive Control of Electrical Drives

Field-oriented control (FOC) has proven effective for controlling ac drives with good dynamic performance. However, operation at low-switching frequencies and the sensitivity of traditional feedforward loops to system parameters pose severe limitations on the achievable performance and require a tedious tuning procedure. In this paper, we present a systematic cascade explicit model predictive control framework for the FOC of electrical drives, resolving the aforementioned issues while being sufficiently simple to be widely implemented on various ac drive systems. The resulting closed-loop system exhibits high dynamic performance for all operating points, even at low-switching frequencies. We present experiments with a permanent-magnet machine and an induction motor, demonstrating the practical feasibility and the merits of the proposed framework over traditional controller designs for electrical drives.

Sensorless explicit model predictive control of permanent magnet synchronous motors

The present paper deals with sensorless model predictive control of permanent magnet synchronous motors. The proposed explicit controllers consist in precomputed (optimal) state feedbacks that are selected according to the measured state using binary search-trees. This type of controller is well suited to obtain very fast control algorithms that impose high dynamic performance. The control scheme is based on two cascaded explicit model predictive controllers, one for the torque control another for the speed control. The nonlinearities of the motor dynamics that are due to the speed are taken into account into the torque controller derivation. The resulting adaptive closed-loop system presents a high dynamic performance for all operating points. A rotor speed and position observer is used to orient the control variables on the rotor flux without position sensor. The overall control scheme allows optimal use of the inverter and drive dynamic capabilities. The proposed concepts are validated experimentally.

Optimization‐based autonomous racing of 1:43 scale RC cars

Summary This paper describes autonomous racing of RC race cars based on mathematical optimization. Using a dynamical model of the vehicle, control inputs are computed by receding horizon based controllers, where the objective is to maximize progress on the track subject to the requirement of staying on the track and avoiding opponents. Two different control formulations are presented. The first controller employs a two-level structure, consisting of a path planner and a nonlinear model predictive controller (NMPC) for tracking. The second controller combines both tasks in one nonlinear optimization problem (NLP) following the ideas of contouring control. Linear time varying models obtained by linearization are used to build local approximations of the control NLPs in the form of convex quadratic programs (QPs) at each sampling time. The resulting QPs have a typical MPC structure and can be solved in the range of milliseconds by recent structure exploiting solvers, which is key to the real-time feasibility of the overall control scheme. Obstacle avoidance is incorporated by means of a high-level corridor planner based on dynamic programming, which generates convex constraints for the controllers according to the current position of opponents and the track layout. The control performance is investigated experimentally using 1:43 scale RC race cars, driven at speeds of more than 3 m/s and in operating regions with saturated rear tire forces (drifting). The algorithms run at 50 Hz sampling rate on embedded computing platforms, demonstrating the real-time feasibility and high performance of optimization-based approaches for autonomous racing. Copyright

On real-time robust model predictive control

High-speed applications impose a hard real-time constraint on the solution of a model predictive control (MPC) problem, which generally prevents the computation of the optimal control input. As a result, in most MPC implementations guarantees on feasibility and stability are sacrificed in order to achieve a real-time setting. In this paper we develop a real-time MPC approach for linear systems that provides these guarantees for arbitrary time constraints, allowing one to trade off computation time vs. performance. Stability is guaranteed by means of a constraint, enforcing that the resulting suboptimal MPC cost is a Lyapunov function. The key is then to guarantee feasibility in real-time, which is achieved by the proposed algorithm through a warm-starting technique in combination with robust MPC design. We address both regulation and tracking of piecewise constant references. As a main contribution of this paper, a new warm-start procedure together with a Lyapunov function for real-time tracking is presented. In addition to providing strong theoretical guarantees, the proposed method can be implemented at high sampling rates. Simulation examples demonstrate the effectiveness of the real-time scheme and show that computation times in the millisecond range can be achieved.

Auto-generated algorithms for nonlinear model predictive control on long and on short horizons

We present a code generation strategy for handling long prediction horizons in the context of real-time nonlinear model predictive control (NMPC). Existing implementations of fast NMPC algorithms use the real-time iteration (RTI) scheme and a condensing technique to reduce the number of optimization variables. Condensing results in a much smaller, but dense quadratic program (QP) to be solved at every time step. While this approach is well suited for short horizons, it leads to unnecessarily long execution times for problem formulations with long horizon. This paper presents a new implementation of auto-generated NMPC code based on a structure exploiting auto-generated QP solver. Utilizing such a QP solver, the condensing step can be avoided and execution times scale linearly with the horizon length instead of cubically. Our simulation results show that this approach significantly decreases the execution time of NMPC with long horizons. For a nonlinear test problem that comprises 9 states and 3 controls on a horizon with 50 time steps, an improvement by a factor of 2 was observed, reducing the execution time for one RTI to below 4 milliseconds on a 3 GHz CPU.

Learning a feasible and stabilizing explicit model predictive control law by robust optimization

Fast model predictive control on embedded systems has been successfully applied to plants with microsecond sampling times employing a precomputed state-to-input map. However, the complexity of this so-called explicit MPC can be prohibitive even for low-dimensional systems. In this paper, we introduce a new synthesis method for low-complexity suboptimal MPC controllers based on function approximation from randomly chosen point-wise sample values. In addition to standard machine learning algorithms formulated as convex programs, we provide sufficient conditions on the learning algorithm in the form of tractable convex constraints that guarantee input and state constraint satisfaction, recursive feasibility and stability of the closed loop system. The resulting control law can be fully parallelized, which renders the approach particularly suitable for highly concurrent embedded platforms such as FPGAs. A numerical example shows the effectiveness of the proposed method.

A model predictive control scheme with torque ripple mitigation for permanent magnet motors

This paper analyses the effect of torque ripple on back-EMF based speed observers for permanent magnet motors. It introduces a scheme that allows to improve these estimates and to identify the motor inherent torque ripple. A high-dynamic performance sensorless model predictive controller is proposed to effectively mitigate selected torque harmonics. The presented concepts are validated experimentally.

Real-Time Motion Planning for Aerial Videography With Real-Time With Dynamic Obstacle Avoidance and Viewpoint Optimization

We propose a method for real-time trajectory generation with applications in aerial videography. Taking framing objectives, such as position of targets in the image plane, as input, our method solves for robot trajectories and gimbal controls automatically and adapts plans in real time due to changes in the environment. We contribute a real-time receding horizon planner that autonomously records scenes with moving targets, while optimizing for visibility under occlusion and ensuring collision-free trajectories. A modular cost function, based on the reprojection error of targets, is proposed that allows for flexibility and artistic freedom and is well behaved under numerical optimization. We formulate the minimization problem under constraints as a finite horizon optimal control problem that fulfills aesthetic objectives, adheres to nonlinear model constraints of the filming robot and collision constraints with static and dynamic obstacles and can be solved in real time. We demonstrate the robustness and efficiency of the method with a number of challenging shots filmed in dynamic environments including those with moving obstacles and shots with multiple targets to be filmed simultaneously.

Power oscillation damping control using wide-area signals: A case study on Nordic equivalent system

Power oscillation damping (POD) control employing wide-area signals is illustrated in an equivalent system model representing key characteristics of Nordic power system. Phasor measurement units (PMUs) in Norway and Finland are used to obtain feedback signals for supplementary control of a large SVC unit located in the south-east of Norway. Comparison has been made between two control design approaches- (i) robust linear time invariant model based POD (MBPOD) - dependant on accurate system model and (ii) indirect adaptive POD (IAPOD) - which is fixed structure but time-varying and relies only on measurements. An optimization problem is formulated to design the controller parameters for MBPOD while the IAPOD is based on online Kalman filter estimation and adaptive pole-shifting control. Performances of both MBPOD and IAPOD are found to be quite similar even though IAPOD requires very little prior information about the system. A number of simulations are carried out under different tie-line outages to verify the performance.